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arXiv · 2602.05697

$L^q$-norm bounds for arithmetic eigenfunctions via microlocal Kakeya-Nikodym estimate

Jiaqi Hou·Xiaoqi Huang
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Abstract

Let $X$ be a compact arithmetic congruence hyperbolic surface, and let $\psi$ be an $L^2$-normalized Hecke-Maass form on $X$ with sufficiently large spectral parameter $\lambda$. We give a new proof to obtain a power saving for the global $L^6$-norm $\|\psi\|_{L^6(X)}\lesssim_\varepsilon\lambda^{\frac{5}{36}+\varepsilon}$ over the local bound $\|\psi\|_{L^6(X)}\lesssim\lambda^{\frac{1}{6}}$ of Sogge. Our method uses a microlocal decomposition for $\psi$ and reduces the $L^6$-norm problem to microlocal Kakeya-Nikodym estimates for $\psi$, and we establish improved microlocal Kakeya-Nikodym estimates via arithmetic amplification developed by Iwaniec and Sarnak.

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Jiaqi Hou, Xiaoqi Huang. 2026-02-05. $L^q$-norm bounds for arithmetic eigenfunctions via microlocal Kakeya-Nikodym estimate. https://arxiv.org/abs/2602.05697

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